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Properties

Label 27.8.a.a.1.1
Level $27$
Weight $8$
Character 27.1
Self dual yes
Analytic conductor $8.434$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,8,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 27.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-128.000 q^{4} +1763.00 q^{7} +12605.0 q^{13} +16384.0 q^{16} +14357.0 q^{19} -78125.0 q^{25} -225664. q^{28} +178916. q^{31} -615373. q^{37} +1.03522e6 q^{43} +2.28463e6 q^{49} -1.61344e6 q^{52} +1.53720e6 q^{61} -2.09715e6 q^{64} -4.05845e6 q^{67} +1.23681e6 q^{73} -1.83770e6 q^{76} -4.24543e6 q^{79} +2.22226e7 q^{91} +5.27636e6 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −128.000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1763.00 1.94272 0.971358 − 0.237622i \(-0.0763680\pi\)
0.971358 + 0.237622i \(0.0763680\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 12605.0 1.59126 0.795630 − 0.605783i \(-0.207140\pi\)
0.795630 + 0.605783i \(0.207140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16384.0 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 14357.0 0.480204 0.240102 − 0.970748i \(-0.422819\pi\)
0.240102 + 0.970748i \(0.422819\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −78125.0 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −225664. −1.94272
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 178916. 1.07866 0.539328 − 0.842096i \(-0.318678\pi\)
0.539328 + 0.842096i \(0.318678\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −615373. −1.99725 −0.998625 − 0.0524236i \(-0.983305\pi\)
−0.998625 + 0.0524236i \(0.983305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.03522e6 1.98561 0.992807 − 0.119727i \(-0.0382021\pi\)
0.992807 + 0.119727i \(0.0382021\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.28463e6 2.77414
\(50\) 0 0
\(51\) 0 0
\(52\) −1.61344e6 −1.59126
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.53720e6 0.867113 0.433556 − 0.901126i \(-0.357258\pi\)
0.433556 + 0.901126i \(0.357258\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.09715e6 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.05845e6 −1.64854 −0.824269 − 0.566198i \(-0.808414\pi\)
−0.824269 + 0.566198i \(0.808414\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.23681e6 0.372111 0.186056 − 0.982539i \(-0.440430\pi\)
0.186056 + 0.982539i \(0.440430\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.83770e6 −0.480204
\(77\) 0 0
\(78\) 0 0
\(79\) −4.24543e6 −0.968783 −0.484392 − 0.874851i \(-0.660959\pi\)
−0.484392 + 0.874851i \(0.660959\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.22226e7 3.09137
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.27636e6 0.586993 0.293497 − 0.955960i \(-0.405181\pi\)
0.293497 + 0.955960i \(0.405181\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e7 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −2.19197e7 −1.97654 −0.988268 − 0.152727i \(-0.951194\pi\)
−0.988268 + 0.152727i \(0.951194\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −1.68240e7 −1.24433 −0.622167 − 0.782884i \(-0.713748\pi\)
−0.622167 + 0.782884i \(0.713748\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.88850e7 1.94272
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.94872e7 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −2.29012e7 −1.07866
\(125\) 0 0
\(126\) 0 0
\(127\) −4.51256e7 −1.95484 −0.977418 − 0.211317i \(-0.932225\pi\)
−0.977418 + 0.211317i \(0.932225\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.53114e7 0.932900
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 979547. 0.0309367 0.0154683 − 0.999880i \(-0.495076\pi\)
0.0154683 + 0.999880i \(0.495076\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 7.87677e7 1.99725
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −8.21144e6 −0.194089 −0.0970443 − 0.995280i \(-0.530939\pi\)
−0.0970443 + 0.995280i \(0.530939\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.20114e7 1.69132 0.845659 − 0.533723i \(-0.179208\pi\)
0.845659 + 0.533723i \(0.179208\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.50255e7 −0.271752 −0.135876 − 0.990726i \(-0.543385\pi\)
−0.135876 + 0.990726i \(0.543385\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 9.61375e7 1.53211
\(170\) 0 0
\(171\) 0 0
\(172\) −1.32509e8 −1.98561
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −1.37734e8 −1.94272
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.53710e8 1.92676 0.963378 − 0.268146i \(-0.0864109\pi\)
0.963378 + 0.268146i \(0.0864109\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −1.03192e8 −1.03323 −0.516613 − 0.856219i \(-0.672807\pi\)
−0.516613 + 0.856219i \(0.672807\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.92432e8 −2.77414
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −7.37380e7 −0.663293 −0.331646 − 0.943404i \(-0.607604\pi\)
−0.331646 + 0.943404i \(0.607604\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.06520e8 1.59126
\(209\) 0 0
\(210\) 0 0
\(211\) −2.88780e7 −0.211630 −0.105815 − 0.994386i \(-0.533745\pi\)
−0.105815 + 0.994386i \(0.533745\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.15429e8 2.09552
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.62897e8 1.58752 0.793759 − 0.608233i \(-0.208121\pi\)
0.793759 + 0.608233i \(0.208121\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −2.01186e8 −1.10707 −0.553534 − 0.832827i \(-0.686721\pi\)
−0.553534 + 0.832827i \(0.686721\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 3.44262e8 1.58427 0.792136 − 0.610345i \(-0.208969\pi\)
0.792136 + 0.610345i \(0.208969\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.96761e8 −0.867113
\(245\) 0 0
\(246\) 0 0
\(247\) 1.80970e8 0.764130
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.68435e8 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −1.08490e9 −3.88009
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 5.19482e8 1.64854
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −6.23847e8 −1.90408 −0.952041 − 0.305969i \(-0.901020\pi\)
−0.952041 + 0.305969i \(0.901020\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.85096e6 0.0165405 0.00827024 − 0.999966i \(-0.497367\pi\)
0.00827024 + 0.999966i \(0.497367\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −4.47604e8 −1.17393 −0.586964 − 0.809613i \(-0.699677\pi\)
−0.586964 + 0.809613i \(0.699677\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.10339e8 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.58312e8 −0.372111
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.82510e9 3.85748
\(302\) 0 0
\(303\) 0 0
\(304\) 2.35225e8 0.480204
\(305\) 0 0
\(306\) 0 0
\(307\) −1.79003e8 −0.353082 −0.176541 − 0.984293i \(-0.556491\pi\)
−0.176541 + 0.984293i \(0.556491\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 5.57008e8 1.02673 0.513365 − 0.858170i \(-0.328399\pi\)
0.513365 + 0.858170i \(0.328399\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.43415e8 0.968783
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −9.84766e8 −1.59126
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.52321e8 0.382434 0.191217 − 0.981548i \(-0.438757\pi\)
0.191217 + 0.981548i \(0.438757\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.14809e9 −1.63408 −0.817039 − 0.576582i \(-0.804386\pi\)
−0.817039 + 0.576582i \(0.804386\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.57589e9 3.44665
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −8.89081e8 −1.11957 −0.559786 − 0.828637i \(-0.689117\pi\)
−0.559786 + 0.828637i \(0.689117\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −6.87748e8 −0.769404
\(362\) 0 0
\(363\) 0 0
\(364\) −2.84449e9 −3.09137
\(365\) 0 0
\(366\) 0 0
\(367\) −1.85289e9 −1.95667 −0.978336 − 0.207021i \(-0.933623\pi\)
−0.978336 + 0.207021i \(0.933623\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.34982e9 1.34678 0.673389 − 0.739288i \(-0.264838\pi\)
0.673389 + 0.739288i \(0.264838\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.28135e8 0.781383 0.390692 − 0.920522i \(-0.372236\pi\)
0.390692 + 0.920522i \(0.372236\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −6.75374e8 −0.586993
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.90824e9 1.53062 0.765308 − 0.643665i \(-0.222587\pi\)
0.765308 + 0.643665i \(0.222587\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.28000e9 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 2.25524e9 1.71642
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.69220e8 −0.339113 −0.169557 − 0.985520i \(-0.554234\pi\)
−0.169557 + 0.985520i \(0.554234\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.80573e9 1.97654
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.92297e8 −0.452173 −0.226087 − 0.974107i \(-0.572593\pi\)
−0.226087 + 0.974107i \(0.572593\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.71008e9 1.68455
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −1.11566e9 −0.660424 −0.330212 − 0.943907i \(-0.607120\pi\)
−0.330212 + 0.943907i \(0.607120\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.15347e9 1.24433
\(437\) 0 0
\(438\) 0 0
\(439\) −3.25154e9 −1.83427 −0.917135 − 0.398576i \(-0.869505\pi\)
−0.917135 + 0.398576i \(0.869505\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −3.69728e9 −1.94272
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.06709e9 1.99332 0.996661 − 0.0816509i \(-0.0260192\pi\)
0.996661 + 0.0816509i \(0.0260192\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 3.08012e9 1.44223 0.721114 − 0.692817i \(-0.243631\pi\)
0.721114 + 0.692817i \(0.243631\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −7.15506e9 −3.20264
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.12164e9 −0.480204
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −7.75678e9 −3.17814
\(482\) 0 0
\(483\) 0 0
\(484\) 2.49436e9 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −4.48049e9 −1.75782 −0.878910 − 0.476988i \(-0.841728\pi\)
−0.878910 + 0.476988i \(0.841728\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 2.93136e9 1.07866
\(497\) 0 0
\(498\) 0 0
\(499\) 3.55148e9 1.27955 0.639775 − 0.768562i \(-0.279028\pi\)
0.639775 + 0.768562i \(0.279028\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 5.77608e9 1.95484
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 2.18049e9 0.722906
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 4.97447e9 1.52052 0.760258 − 0.649621i \(-0.225072\pi\)
0.760258 + 0.649621i \(0.225072\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.40483e9 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −3.23986e9 −0.932900
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.36581e9 2.00000 1.00000 0.000770940i \(-0.000245398\pi\)
1.00000 0.000770940i \(0.000245398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.14149e9 0.298205 0.149103 − 0.988822i \(-0.452362\pi\)
0.149103 + 0.988822i \(0.452362\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.48469e9 −1.88207
\(554\) 0 0
\(555\) 0 0
\(556\) −1.25382e8 −0.0309367
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 1.30490e10 3.15963
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −8.66636e9 −1.94810 −0.974048 − 0.226342i \(-0.927323\pi\)
−0.974048 + 0.226342i \(0.927323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −9.21890e9 −1.99785 −0.998927 − 0.0463088i \(-0.985254\pi\)
−0.998927 + 0.0463088i \(0.985254\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 2.56870e9 0.517976
\(590\) 0 0
\(591\) 0 0
\(592\) −1.00823e10 −1.99725
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 7.41654e9 1.39361 0.696804 − 0.717262i \(-0.254605\pi\)
0.696804 + 0.717262i \(0.254605\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.05106e9 0.194089
\(605\) 0 0
\(606\) 0 0
\(607\) −8.14148e9 −1.47755 −0.738776 − 0.673951i \(-0.764596\pi\)
−0.738776 + 0.673951i \(0.764596\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7.09176e9 −1.24349 −0.621745 − 0.783220i \(-0.713576\pi\)
−0.621745 + 0.783220i \(0.713576\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −7.61134e9 −1.28986 −0.644932 − 0.764240i \(-0.723114\pi\)
−0.644932 + 0.764240i \(0.723114\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.10352e9 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.04975e10 −1.69132
\(629\) 0 0
\(630\) 0 0
\(631\) 9.94859e9 1.57637 0.788186 − 0.615437i \(-0.211021\pi\)
0.788186 + 0.615437i \(0.211021\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.87977e10 4.41438
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.04945e9 0.155677 0.0778386 − 0.996966i \(-0.475198\pi\)
0.0778386 + 0.996966i \(0.475198\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.92327e9 0.271752
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 8.20348e9 1.10482 0.552412 − 0.833571i \(-0.313708\pi\)
0.552412 + 0.833571i \(0.313708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.04948e10 −1.32715 −0.663576 − 0.748109i \(-0.730962\pi\)
−0.663576 + 0.748109i \(0.730962\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.23056e10 −1.53211
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 9.30222e9 1.14036
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.69611e10 1.98561
\(689\) 0 0
\(690\) 0 0
\(691\) −1.52104e10 −1.75375 −0.876876 − 0.480717i \(-0.840376\pi\)
−0.876876 + 0.480717i \(0.840376\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.76300e10 1.94272
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −8.83491e9 −0.959088
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.47091e10 1.54998 0.774989 − 0.631974i \(-0.217755\pi\)
0.774989 + 0.631974i \(0.217755\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −3.86445e10 −3.83985
\(722\) 0 0
\(723\) 0 0
\(724\) −1.96749e10 −1.92676
\(725\) 0 0
\(726\) 0 0
\(727\) −8.01595e9 −0.773722 −0.386861 − 0.922138i \(-0.626441\pi\)
−0.386861 + 0.922138i \(0.626441\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.97732e10 −1.85445 −0.927223 − 0.374511i \(-0.877811\pi\)
−0.927223 + 0.374511i \(0.877811\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.90404e10 1.73549 0.867743 − 0.497013i \(-0.165570\pi\)
0.867743 + 0.497013i \(0.165570\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 8.33632e9 0.718182 0.359091 − 0.933302i \(-0.383087\pi\)
0.359091 + 0.933302i \(0.383087\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.61656e10 1.35443 0.677213 − 0.735787i \(-0.263187\pi\)
0.677213 + 0.735787i \(0.263187\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −2.96608e10 −2.41739
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.43003e10 1.92694 0.963472 − 0.267808i \(-0.0862994\pi\)
0.963472 + 0.267808i \(0.0862994\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.32086e10 1.03323
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −1.39778e10 −1.07866
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.74313e10 2.77414
\(785\) 0 0
\(786\) 0 0
\(787\) −2.23421e10 −1.63385 −0.816926 − 0.576743i \(-0.804323\pi\)
−0.816926 + 0.576743i \(0.804323\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.93764e10 1.37980
\(794\) 0 0
\(795\) 0 0
\(796\) 9.43846e9 0.663293
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 8.58179e9 0.564944 0.282472 − 0.959276i \(-0.408846\pi\)
0.282472 + 0.959276i \(0.408846\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.48627e10 0.953500
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −1.83431e10 −1.14702 −0.573512 − 0.819197i \(-0.694419\pi\)
−0.573512 + 0.819197i \(0.694419\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 2.47130e10 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.64346e10 −1.59126
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −1.72499e10 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 3.69638e9 0.211630
\(845\) 0 0
\(846\) 0 0
\(847\) −3.43559e10 −1.94272
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.54191e10 1.95396 0.976981 − 0.213326i \(-0.0684297\pi\)
0.976981 + 0.213326i \(0.0684297\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 3.71535e10 1.99997 0.999986 − 0.00533998i \(-0.00169978\pi\)
0.999986 + 0.00533998i \(0.00169978\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −4.03749e10 −2.09552
\(869\) 0 0
\(870\) 0 0
\(871\) −5.11568e10 −2.62325
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.26331e10 1.63366 0.816828 − 0.576882i \(-0.195731\pi\)
0.816828 + 0.576882i \(0.195731\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −4.16837e9 −0.203753 −0.101876 − 0.994797i \(-0.532485\pi\)
−0.101876 + 0.994797i \(0.532485\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −7.95565e10 −3.79769
\(890\) 0 0
\(891\) 0 0
\(892\) −3.36508e10 −1.58752
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.50682e10 1.56059 0.780293 − 0.625415i \(-0.215070\pi\)
0.780293 + 0.625415i \(0.215070\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 2.57518e10 1.10707
\(917\) 0 0
\(918\) 0 0
\(919\) 3.77715e10 1.60532 0.802658 − 0.596440i \(-0.203419\pi\)
0.802658 + 0.596440i \(0.203419\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.80760e10 1.99725
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 3.28004e10 1.33216
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.17131e10 −1.65647 −0.828236 − 0.560379i \(-0.810655\pi\)
−0.828236 + 0.560379i \(0.810655\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 1.55900e10 0.592125
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.49832e9 0.163500
\(962\) 0 0
\(963\) 0 0
\(964\) −4.40656e10 −1.58427
\(965\) 0 0
\(966\) 0 0
\(967\) 5.32711e10 1.89452 0.947260 − 0.320467i \(-0.103840\pi\)
0.947260 + 0.320467i \(0.103840\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 1.72694e9 0.0601011
\(974\) 0 0
\(975\) 0 0
\(976\) 2.51855e10 0.867113
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.31642e10 −0.764130
\(989\) 0 0
\(990\) 0 0
\(991\) 1.10040e10 0.359163 0.179581 − 0.983743i \(-0.442526\pi\)
0.179581 + 0.983743i \(0.442526\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.61437e10 1.79419 0.897093 − 0.441841i \(-0.145674\pi\)
0.897093 + 0.441841i \(0.145674\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 27.8.a.a.1.1 ✓ 1
3.2 odd 2 CM 27.8.a.a.1.1 ✓ 1
4.3 odd 2 432.8.a.d.1.1 1
9.2 odd 6 81.8.c.b.28.1 2
9.4 even 3 81.8.c.b.55.1 2
9.5 odd 6 81.8.c.b.55.1 2
9.7 even 3 81.8.c.b.28.1 2
12.11 even 2 432.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
27.8.a.a.1.1 ✓ 1 3.2 odd 2 CM
81.8.c.b.28.1 2 9.2 odd 6
81.8.c.b.28.1 2 9.7 even 3
81.8.c.b.55.1 2 9.4 even 3
81.8.c.b.55.1 2 9.5 odd 6
432.8.a.d.1.1 1 4.3 odd 2
432.8.a.d.1.1 1 12.11 even 2